In a stand mixer fitted with the paddle attachment, mix the butter and sugar together until just incorporated. Do not over-mix at this stage, or the cookies may spread while baking.
Add the egg and vanilla and almond extracts. Mix again on low speed, stopping to scrape down the sides of the bowl intermittently as needed.
In a medium bowl, whisk together the flour and salt. Add to the butter and egg mixture. Mix on low speed until a dough is formed and there are no longer any streaks of butter in the mixing bowl. The dough will often clump around the paddle attachment while being mixed. This is normal and a good sign that your dough is the right consistency. If your mixture does not come together and is crumbly, add ice cold water 1 tbsp. at a time until the dough clumps.
Divide the dough in half and return one half to the mixing bowl. Add the red food color and mix well until a deep red color is achieved.
Divide the red and white dough in half so that you have four dough portions. Form each portion into a rectangle and wrap individually in plastic; refrigerate until well chilled.
Remove one red and one white dough portion from the refrigerator. Roll each rectangle to about 1/2-inch thickness, and then cut them each into nine long 1/2-inch thick square strips.
Gently stack the strips forming two checkerboard logs. Cover with plastic wrap or parchment paper and refrigerate the logs until firm, about 2 hours.
Roll each remaining dough piece out to about 1/4 inch thickness. Wrap the chilled dough logs in the flat dough in a single layer. Cut away any excess dough. Cover in plastic wrap and chill again, about 30 minutes.
Preheat the oven to 350°F.
Cover 2 or more cookie sheets with parchment paper. Slice each dough log into squares about 1/4 inch thick and place them on the prepared baking sheets. Bake the cookies for 12 to 15 minutes, or until lightly golden on the bottoms. Let cool on the baking sheet for 5 minutes. Transfer to a wire rack. Decorate the cookies when completely cooled.
Make the peppermint buttercream:
Combine the butter and confectioners’ sugar in a large bowl. Beat on low speed until just combined; add the extract. Beat on high speed adding milk or cream 1 tablespoon at a time until the icing is of piping consistency. Beat on high speed for 3 to 5 minutes, or until the frosting is light in color and fluffy.Generously pipe or spread frosting on cookies and top with a second cookie.
Package two or three sandwich cookies in cellophane bags and tie with festive ribbon. Add a gift tag and give to delighted friends and loved ones.